\section{Disturbance torques on a spacecraft}

\begin{frame}{\thesection.\ \insertsection}
There are a number of external torques acting on a spacecraft which disturb the attitude motion. \\
\vfill
For a spacecraft in the vicinity of the Earth, the major disturbance torques are:
\begin{enumerate}
    \item Magnetic torque
    \item Solar radiation pressure torque
    \item Aerodynamic torque
    \item Gravity-gradient torque
\end{enumerate}
\vfill
\end{frame}

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\subsection{Magnetic torque}
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\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The cause of the magnetic torque is the interaction between the Earth’s magnetic field and any magnetization of the spacecraft (electronic components create an equivalent current loop, which results in a magnetic dipole).  
\vfill
The torque on the spacecraft due to the interaction of the Earth’s magnetic field \(\vec{b}\) and the spacecraft residual magnetic dipole moment \(\vec{_{}m}\) is given by  
\[\vec{_{}T}_m = \vec{_{}m} \times \vec{b}\]
\vfill
\end{frame}

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\subsection{Solar radiation pressure torque}
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\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The output from the sun (photons) contain momentum, which produce an effective pressure on spacecraft surfaces (due to momentum transfer).
\begin{itemize}
    \item There are different modes of interaction between the solar radiation and the spacecraft surface, which depends on the spacecraft surface properties.
    \item In practice, all three types of interaction are present in different proportions.
\end{itemize}
\vspace{-6pt}
\begin{center}\includegraphics[scale=0.5]{fig_8_1.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.1:} Interaction of radiation with surface\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Assume total absorption. \\
The total solar pressure torque is given by
\[\vec{_{}T}_s = \vec{c}_{ps} \times \vec{F}_s\]
where
\[\vec{c}_{ps} = \frac{\int_{S_{ws}} \vec{p}(\vec{n} \cdot \vec{s}) \text dS}{\int_{S_{ws}} (\vec{n} \cdot \vec{s}) \text dS},
\quad \vec{F}_s = -p\vec{s} \int_{S_{ws}} \vec{n} \cdot \vec{s} \text dS, \quad p = 4.5 \times 10^{-6} \, \text{N/m}^2\]  
\(\vec{n}\) is the unit outward normal, \(\vec{s}\) is the sun's direction, \(\vec{p}\) is the location of \(dS\) from the spacecraft center of mass, \(S_{ws}\) is the wetted (lit) portion of the spacecraft surface (the portion for which \(\vec{n} \cdot \vec{s} \geq 0\)).
\vspace{-4pt}
\begin{center}\includegraphics{fig_8_2.pdf}\end{center}
\vspace{-8pt}
\begin{center}\textcolor{blue}{Figure \arabic{section}.2:} Solar radiation pressure force on a surface element\end{center}
\end{frame}

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\subsection{Aerodynamic torque}
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\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
In low Earth orbits, there is still some residual atmosphere. \\
The aerodynamic torque is given by
\[\vec{_{}T}_a = \vec{c}_{pa} \times \vec{F}_a\]
\vspace{-9pt}
where
\vspace{-3pt}
\[\vec{c}_{pa} = \frac{\int_{S_{wa}} \vec{\rho}(\vec{n} \cdot \hat{\vec{v}}) dS}{\int_{S_{wa}} (\vec{n} \cdot \hat{\vec{v}}) dS}, \quad \vec{F}_a = -\rho_a v^2 \hat{\vec{v}} \int_{S_{wa}} \vec{n} \cdot \hat{\vec{v}} dS\]  
\(\vec{n}\) is the unit outward normal, \(\vec{v}\) is the orbital velocity of the spacecraft, \(v = |\vec{v}|\), \(\hat{\vec{v}} = \frac{\vec{v}}{v}\), \(\vec{\rho}\) is the location of \(dS\) from the spacecraft center of mass, \(S_{wa}\) is the wetted area (area facing the flow, given by \(\vec{n} \cdot \hat{\vec{v}} \geq 0\)), \(\rho_a\) is the atmospheric density.
\vspace{-12pt}
\begin{center}\includegraphics[scale=0.8]{fig_8_3.pdf}\end{center}
\vspace{-12pt}
\begin{center}\textcolor{blue}{Figure \arabic{section}.3:} Aerodynamic force on a surface element\end{center}
\end{frame}

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\subsection{Gravity-gradient torque}
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\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The gravity-gradient torque is due to the fact that the Earth’s gravitational force is not constant with distance from the Earth’s center but decreases quadratically.
\begin{columns}
\column{0.4\textwidth}
\begin{center}\includegraphics{fig_8_4.pdf}\end{center}
\textcolor{blue}{Figure \arabic{section}.4:} Gravity-gradient torque concept
\column{0.5\textwidth}
\begin{itemize}
    \item The gravitational force on a mass further from the Earth is smaller than the force on a mass that is closer.
    \item This gravity-gradient produces a torque.
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
The gravity-gradient torque is given by
\[ T_g = \frac{3\mu}{r^5}r_b^\times Ir_b \]
where \(\vec{r}\) is the orbital position of the spacecraft center of mass, \(r = |\vec{r}|\), \(r_b\) is the spacecraft orbital position in spacecraft body coordinates, \(I\) is the moment of inertia matrix.
\begin{center}\includegraphics{fig_8_5.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.5:} Modeling gravity-gradient torque for a continuous body\end{center}
\end{frame}
